Relativistic Euler equations

# Relativistic Euler equations

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In fluid mechanics
Fluid mechanics
Fluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...

and astrophysics
Astrophysics
Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties of celestial objects, as well as their interactions and behavior...

, the relativistic Euler equations are a generalization of the Euler equations that account for the effects of special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

.

The equations of motion are contained in the continuity equation
Continuity equation
A continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...

of the stress-energy tensor
Stress-energy tensor
The stressâ€“energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...

:

where the right hand side is the zero tensor. For a perfect fluid
Perfect fluid
In physics, a perfect fluid is a fluid that can be completely characterized by its rest frame energy density ρ and isotropic pressure p....

,
.

Here is the relativistic rest energy density of the fluid, is the fluid pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

, is the four-velocity
Four-velocity
In physics, in particular in special relativity and general relativity, the four-velocity of an object is a four-vector that replaces classicalvelocity...

of the fluid, and is the Minkowski metric tensor.

To the above equations, a statement of conservation
Conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....

is usually added, usually conservation of baryon number. If is the number density of baryon
Baryon
A baryon is a composite particle made up of three quarks . Baryons and mesons belong to the hadron family, which are the quark-based particles...

s this may be stated

These equations reduce to the classical Euler equations if .

The relativistic Euler equations may be applied to calculate the speed of sound
Speed of sound
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....

in a fluid with a relativistic equation of state
Equation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...

(that is, one in which the pressure is comparable with the internal energy
Internal energy
In thermodynamics, the internal energy is the total energy contained by a thermodynamic system. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields. Internal...

density , including the rest energy; where is the classical internal energy per unit mass).

Under these circumstances, the speed of sound is given by

(note that

is the relativistic internal energy density). This formula differs from the classical case in that has been replaced by .